Theorem cocanfo 37916

Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
cocanfo	⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹))
2	1	fveq1d 6836	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = ((𝐻 ∘ 𝐹)‘𝑦))
3		simpl1 1192	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴–onto→𝐵)
4		fof 6746	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴⟶𝐵)
6		fvco3 6933	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
7	5, 6	sylan 580	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
8		fvco3 6933	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
9	5, 8	sylan 580	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
10	2, 7, 9	3eqtr3d 2779	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
11	10	ralrimiva 3128	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
12		fveq2 6834	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐺‘(𝐹‘𝑦)) = (𝐺‘𝑥))
13		fveq2 6834	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
14	12, 13	eqeq12d 2752	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ (𝐺‘𝑥) = (𝐻‘𝑥)))
15	14	cbvfo 7235	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
16	3, 15	syl 17	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
17	11, 16	mpbid 232	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥))
18		eqfnfv 6976	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
19	18	3adant1 1130	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
20	19	adantr 480	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
21	17, 20	mpbird 257	1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500

This theorem is referenced by: (None)